Results Related to the LDP: An Explanation

Abstract and 1 Introduction

1.1 State of the art

1.2 Some remarks on dynamics and initial condition

1.3 Outline of the paper

1.4 List of notations

2 Large Deviation Principle

2.1 Establishing the LDP for the SID

2.2 Results related to the LDP

2.3 Compactness results

3 Exit-time

3.1 Auxiliary results

3.2 Proof of the main theorem

3.3 Proofs of auxiliary lemmas

4 Generalization and References

The following lemma generalizes the large deviation principle for the case of converging initial conditions.

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\ and that proves the first inequality. One can prove the second inequality the same way.

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\ As was pointed out before, convergence of measures in Wasserstein distance gives convergence of respective integrals, since ∇F is Lipschitz continuous [Vil09, Theorem 6.9].

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\ As was pointed out before, lower semicontinuity guarantees that infima of a function are achieved over compact sets. We summarise this property by the following corollary.

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:::info This paper is available on arxiv under CC BY-SA 4.0 DEED license.

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:::info Authors:

(1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France;

(2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France;

(3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.

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